2021/10/26 by David Bryant, Katharina T. Huber, Bryant, David +5
Computer Science · Mathematics · #51F99 #52A20 #Advanced Topology and Set Theory #FOS: Mathematics #Metric Geometry (math.MG) #Optimization and Variational Analysis #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2110.13383
openalex publication_date 2021/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The generalized circumradius of a set of points A ⊆ ℝd with respect to a convex body K equals the minimum value of λ≥ 0 such that A is contained in a translate of λK. Each choice of K gives a different function on the set of bounded subsets of ℝd; we characterize which functions can arise in this way. Our characterization draws on the theory of diversities, a recently introduced generalization of metrics from functions on pairs to functions on finite subsets. We additionally investigate functions which arise by restricting the generalised circumradius to a finite subset of ℝd. We obtain elegant characterizations in the case that K is a simplex or parallelotope.